WILLIAMSBURG, Va. — House Republicans have their Tea Party Caucus. They have their G.O.P. Doctors Caucus. And, joining the list of varied special interest caucuses, they recently picked up another influential but much more unofficial group — the Vote No/Hope Yes Caucus.

These are the small but significant number of Republican representatives who, on the recent legislation to head off the broad tax increases and spending cuts mandated by the so-called fiscal cliff, voted no while privately hoping — and at times even lobbying — in favor of the bill’s passage, given the potential harmful economic consequences otherwise.

Representative Tom Cole of Oklahoma, part of the Republican whip team responsible for marshaling support for legislation, said the current makeup of House Republicans could be divided roughly into a third who voted in favor of the bill because they wanted it to pass, a third who voted against the bill because they wanted it to fail, and a third who voted against the bill but had their fingers crossed that it would pass and avert a fiscal and political calamity.

One lawmaker, Mr. Cole said, told him that while he did not want to vote in favor of the bill, he also did not want to amend it and send it back to the Senate where it might die and leave House Republicans blamed for tax increases. “So I said, ‘What you’re really telling me is that you want it to pass, but you don’t want to vote for it,'” recalled Mr. Cole, who voted yes.